
We denote this associative algebra by \(\mathfrak {A}_\mathfrak {R}\) . Moreover, if \((\mathcal {M}, \mathcal {R})\)  is a Rota-Baxter bimodule over the \(\lambda \) -weighted Rota-Baxter associative algebra \((\mathfrak {A}, \mathfrak {R})\) , then it has been observed in [1]} that \(\mathcal {M}\)  carries a bimodule structure over the associative algebra \(\mathfrak {A}_\mathfrak {R}\)  with left and right actions
\(a ~\widetilde{\cdot }~ m = \mathfrak {R}(a) \cdot m - \mathcal {R} (a \cdot m) ~~ \text{ and } ~~ m ~\widetilde{\cdot }~ a = m \cdot \mathfrak {R}(a) - \mathcal {R} ( m \cdot a), ~ \text{ for } a \in \mathfrak {A}_\mathfrak {R} ~\text{ and } m \in \mathcal {M}.\) 
